relations between entanglement and purity in non-markovian … · 2016-04-26 · relations between...

Relations between entanglement and purity innon-Markovian dynamics

Carlos A. Gonzalez-Gutierrez1,2, RicardoRoman-Ancheyta1,2, Diego Espitia2,3 and Rosario LoFranco4

1Posgrado en Ciencias Fısicas, Universidad Nacional Autonoma de Mexico2Instituto de Ciencias Fısicas, Universidad Nacional Autonoma de Mexico,Avenida Universidad s/n, 62210 Cuernavaca, Morelos, Mexico3 Centro de Investigaciones en Ciencias, Universidad Autonoma del Estado deMorelos, 62209 Cuernavaca, Morelos, Mexico4 Dipartimento di Energia, Ingegneria dell’Informazione e Modelli Matematici,Universita di Palermo, Viale delle Scienze, Ed. 9, 90128 Palermo, Italy

E-mail: [email protected], [email protected],

[email protected], [email protected]

Abstract. Knowledge of the relationships among different features ofquantumness, like entanglement and state purity, is important from bothfundamental and practical viewpoints. Yet, this issue remains little exploredin dynamical contexts for open quantum systems. We address this problem bystudying the dynamics of entanglement and purity for two-qubit systems usingparadigmatic models of radiation-matter interaction, with a qubit being isolatedfrom the environment (spectator configuration). We show the effects of thecorresponding local quantum channels on an initial two-qubit pure entangledstate in the concurrence-purity diagram and find the conditions which enabledynamical closed formulas of concurrence, used to quantify entanglement, as afunction of purity. We finally discuss the usefulness of these relations in assessingentanglement and purity thresholds which allow noisy quantum teleportation.Our results provide new insights about how different properties of composite openquantum systems behave and relate each other during quantum evolutions.

1. Introduction

Dynamics of composite quantum systems interacting with their surroundings isof central interest for understanding how quantum features are affected by theenvironment and for controlling them in view of their exploitation as quantuminformation resources [1, 2, 3, 4, 5]. A quantum system made of two qubits is asuitable theoretical platform to analyze entanglement and coherence evolution fromthe perspective of the theory of open quantum systems [6, 7]. The study of this simplesystem is of special relevance as it constitutes the basic building block for quantumgates and quantum teleportation protocols [8, 9] which can be affected by externalundesired interactions.

Physical models in the context of quantum optics have been widely used inthe study of intrinsic decay of quantum coherences for one or more qubits due tothe interaction with the quantized electromagnetic radiation field. For instance,

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Relations between entanglement and purity in non-Markovian dynamics 2

the Jaynes-Cummings model or its generalization to a collection of N two-levelatoms (or qubits) known as the Tavis-Cummings model are typical settings [10, 11].The study of bipartite entanglement between one qubit and the field [12, 13, 14]or of multipartite entanglement among qubits [15, 16], has led to discover manyinteresting phenomena and also to experimental proposals for quantum protocols.Some examples are entanglement in simple quantum phase transitions [17], protocolsfor Bell state measurements [18], physical implementation of quantum gates [8, 19, 20]and generation of quantum correlations in qubit networks [21].

One of the main drawbacks encountered when two qubits locally interactwith their own Markovian (memoryless) environment is the so-called entanglementsudden death (ESD), that is the complete disappearance of entanglement at a finitetime [22, 23, 24, 25, 26, 27, 28]. This phenomenon, whose experimental evidence hasbeen also proved [29, 30], has then motivated the development of efficient strategiesto avoid it [31, 32, 33, 34, 35, 36, 37, 38] or delay it [39, 40, 41, 42, 43, 44, 45],typically based on suitable non-Markovian (memory-keeping) environments and localoperations.

The scope of this paper is to provide new insights about the non-Markoviandynamics of the quantum correlations captured by entanglement from the perspectiveof its interplay with purity, which identifies the degree of mixedness of a quantum statebeing related to coherence. Such a study is still little addressed [46, 47], particularlyin the presence of local non-Markovian environments [48]. Knowledge of relationshipsbetween entanglement and purity in specific dynamical contexts is important not onlyfrom a fundamental viewpoint but also from a practical one. In fact, it would providequantitative thresholds of entanglement for a given purity at a certain time whichallow quantum protocols, like teleportation [49, 50], entanglement swapping [51] andentanglement percolation [52]. To this aim our strategy is to consider a two-qubitcentral system using the so-called spectator configuration [53], where one of the qubitsis isolated and acts as a probe. This idealized configuration is a convenient way toinvestigate non-trivial dynamics of entanglement versus purity for two qubits withoutany type of interaction between them. On the other hand, the characterization ofquantum processes under particular channels or operations within this simple openquantum system can be implemented experimentally. Realizations of unital andnonunital (both Markovian and non-Markovian) channels acting on one of two qubitsare posible using all-optical setups [34, 54, 41, 55] and are also achievable in circuitQED devices [11, 14]. For our analysis we shall also employ the concurrence-purity (C-P ) plane [46], which is a powerful tool that brings a general overview of the systemdynamics and it is not commonly used in quantum optics literature. We focus onthree models which shall allow us to obtain exact analytical results with a consequentbetter understanding of the system evolution, namely: Tavis-Cummings (TC), Buck-Sukumar (BS) and spin-boson (SB) models.

The paper is structured as follows. In Sections 2, 3 and 4 we analyze the two-qubitTC, BS and SB models, respectively, for which we obtain the exact time evolutionfor the reduced density operator of the central system. Exact expressions for purityand concurrence are also derived. In Sec. 5 we discuss the results and explore the C-P diagram identifying the nature of different decoherence processes in the two-qubitcentral system and their use for implementing noisy teleportation. Finally, in Sec. 6we give our conclusions.


2. Two-qubit Tavis-Cummings model

The interaction between two identical two-level atoms (qubits) A and B with a singlemode of the electromagnetic radiation field with frequency ν in the dipole and rotating-wave approximations is described by the following Tavis-Cummings Hamiltonian [10](we set ~ = 1)

HTC =ω0

2(σA

z + σBz ) + νa†a+

∑

j=A,B

gj(aσj

+ + a†σj−), (1)

where a, a† are the usual bosonic operators satisfying [a, a†] = 1, σjz is the z-component

of Pauli matrices and σj± are the rising and lowering operators for atoms A and B.

We remark that this model is experimentally realized in circuit QED [11]. Here wewill focus on the particular case where gB = 0 and gA = g, i.e., only one of theatoms is interacting with the field. This setting could be realized with the atom Boutside of the cavity [56] or with it in a node of the electromagnetic field. In thiscontext the atom B acts as a probe from which one can obtain information aboutthe other systems (atom A and/or the field). This is the spectator configuration.For simplicity, in the following we restrict our analysis to the resonant case ω0 = ν.With the aforementioned considerations it is easy to obtain the exact time evolutionoperator for the Hamiltonian of Eq. (1) in the interaction picture and in the atomicbasis {|ee〉, |eg〉, |ge〉, |gg〉}, which reads

U(t)TC =

(cos(gt

√aa†) −iV sin(gt

√a†a)

−iV † sin(gt√aa†) cos(gt

√a†a)

)⊗ 1B , (2)

where 1B is the identity operator for the qubit B Hilbert space, and we have used thewell known expression for the Jaynes-Cummings (JC) time propagator [13] in termsof the Susskind-Glogower operators defined as [57]

V =1√

a†a+ 1a =

∞∑

n=0

|n〉〈n+ 1|, V † = a†1√

a†a+ 1=

∞∑

n=0

|n+ 1〉〈n|. (3)

These operators are non-unitary and satisfy the commutation relation [V, V †] = |0〉〈0|.In order to investigate the reduced dynamics of the two-qubit system we assume

the total initial state as a product state ρ(0) = %Ψ(0)⊗ ρf (0) where

%Ψ(0) =1− x

41 + x|ψ〉〈ψ|, (4)

is a Werner-like state for the central system with purity parameter x ∈ [0, 1],|ψ〉 = sinφ|ee〉 + cosφ|gg〉 and ρf (0) is an arbitrary initial state of the field. Sucha state reduces to a Bell-like state when x = 1 and is contained in a wider class oftwo-qubit states known as X states, which are represented by a density matrix havingonly diagonal and off-diagonal terms different from zero [7]. We focus on two particularfield states of interest: the number state and coherent state, which represent the mostquantum and the most classical states of the radiation field, respectively.

2.1. Field in a number state

In this case we consider the field to be initially in a pure state with a definite numberof photons, i.e., ρf (0) = |n〉〈n|. Using this field state in ρ(0) and tracing over the


degrees of freedom of the field we can get the reduced density operator for the centralsystem as

%(t) = trf [U(t)ρ(0)U†(t)]. (5)

For simplicity we write down only the non zero matrix elements for the reduced densityoperator

%11(t) =(1− x

4+ x sin2 φ

)cos2(gt

√n+ 1) +

1− x4

sin2(gt√n),

%22(t) =(1− x

4+ x cos2 φ

)sin2(gt

√n) +

1− x4

cos2(gt√n+ 1),

%33(t) =(1− x

4+ x sin2 φ

)sin2(gt

√n+ 1) +

1− x4

cos2(gt√n), (6)

%44(t) =(1− x

4+ x cos2 φ

)cos2(gt

√n) +

1− x4

sin2(gt√n+ 1),

%14(t) = x sinφ cosφ cos(gt√n+ 1) cos(gt

√n), %41(t) = %14(t)

∗.

Notice that the reduced density operator maintains during the time evolution its initialX structure. With the reduced density matrix of Eq. (6) we can calculate at any timethe evolution of purity and concurrence for the central system which are standardmeasurements of decoherence and entanglement.

To quantify the loss of coherence trough the degree of mixedness of the two-qubitsystem we use the purity of a density operator which is defined as

P (t) = tr[%(t)2]. (7)

The purity takes its maximum value of one if the state is a one-dimensional projector,i.e. if it is a pure state. The minimum value of this quantity is bounded by the inverseof the dimension of the system Hilbert space.

The entanglement shared between two qubits can be quantified using theconcurrence, which is defined for a general mixed state % as [58]

C(%) = max{0, λ1 − λ2 − λ3 − λ4}, (8)

where λi are the square roots of the eigenvalues of %% in non-increasing order. Theoperator % is obtained by applying a spin flip operation on %, i.e, % = (σy⊗σy)%∗(σy⊗σy) and the complex conjugate is taken in the atomic basis of the two qubits.

For a X state of the form of Eq. (6), the concurrence can be easily obtained via [46]

C(%X) = 2 max{0, |%14| −√%22%33}. (9)

To get easy to handle explicit expressions of the quantifiers, we analyze the particularcase with x = 1 and φ = π/4, which corresponds to an initial pure Bell state of thetwo-qubit system. A straightforward calculation shows that purity and concurrenceread

P (t) =1

2+

1

8

[4 cos2(gt

√n) cos2(gt

√n+ 1)− 1

]+

1

16

[cos (4gt

√n) + cos (4gt

√n+ 1)

].

(10)

C(t) = 2 max{0, 1

2

(∣∣cos(gt√n) cos(gt

√n+ 1)

∣∣−∣∣sin(gt

√n) sin(gt

√n+ 1)

∣∣)}. (11)


0 π/2 π3π/2 2π 5π/2 0

π/4

π/2

Conc

urre

nce

0

0.2

0.4

0.6

0.8

1.0

gt

φ

(a)

0 π/2 π3π/2 2π 5π/2 0

π/4

π/2

Conc

urre

nce

0.0

0.05

0.10

0.15

0.20

gt

φ

(b)

Figure 1: Concurrence as a function of scaled time gt and initial degree of entanglementφ for the spectator two-qubit TC model in the vacuum state n = 0. Two casesare shown: (a) with purity parameter x = 1 there is vanishing of entanglement atgt = (m+1/2)π, (b) For x = 0.48 collapses and revivals of entanglement are observed.This behavior shows a strong dependence on the initial conditions, as reported inRef. [23].

We notice that for n = 0 (vacuum field state), purity and concurrence are related via

C(t) = 4√

2P (t)− 1, (12)

which is the typical behaviour that characterizes a homogenization process in a C-Pdiagram [46] and tells us that the two qubits are entangled whenever the purity islarger than 1/2. This process belongs to a class of non-unital channels (see Sec. 5 fordetails).

In Fig. 1 we show the evolution of concurrence by substituting the matrix elementsof Eq. (6) in Eq. (9) with the field in the vacuum state n=0 and an arbitrary initialdegree of entanglement. The figure shows two cases: (a) pure state (x = 1), forwhich C(t) = 2 max{0, | cosφ sinφ cos gt|} and (b) mixed state (x = 0.48). The timebehaviors are in accordance with the non-dissipative case of a single qubit subject toa single-mode radiation field in the vacuum state (zero temperature perfect cavity).

2.2. Field in a coherent state

We now choose the initial radiation field in a coherent state, which is a typical situationin cavity-QED experiments [12]. In this case the field state is given by ρf (0)=|α〉〈α|,where |α〉=∑∞m=0 Cm|m〉 with Cm=exp(−|α|2/2)αm/

√m!. The explicit elements of


0 20 40 60 80 100gt

0

0.2

0.4

0.6

0.8

1.0

P(t

)C

(t)

Figure 2: Purity (upper red line) and concurrence (lower dark blue line) as functionsof scaled time gt for the spectator two-qubit TC model. Bell (x = 1, φ = π/4) andcoherent (n = 15) initial states were used. Concurrence shows collapses and revivalsof entanglement with the envelope eventually decaying at gtr ≈ 2π

√n.

the reduced density operator for x = 1 and φ = π/4 are

%11 =∑

m

|Cm|2 cos2(gt√m+ 1), %13 = i

∑

m

C∗m+1Cm sin(gt√m+ 1) cos(gt

√m+ 2),

%12 =i

2

∑

m

C∗m+1Cm sin(2gt√m+ 1), %14 =

∑

m

|Cm|2 cos(gt√m) cos(gt

√m+ 1),

%22 =∑

m

|Cm|2 sin2(gt√m+ 1), %23 =

∑

m

C∗mCm+2 sin(gt√m+ 1) sin(gt

√m+ 2),

%33 =∑

m

|Cm|2 sin2(gt√m), %24 = −i

∑

m

C∗mCm+1 sin(gt√m+ 1) cos(gt

√m),

%44 =∑

m

|Cm|2 cos2(gt√m), %34 = − i

2

∑

m

C∗m+1Cm sin(2gt√m+ 1), (13)

where we have omitted the explicit time dependence in the matrix elements %jk(t). Asin the standard JC model, the sums in Eq. (13) cannot be evaluated in a closed form,so analytical expressions for purity and concurrence are too cumbersome to be shownhere. In Fig. 2 we then show plots of purity and concurrence as functions of time, wherethe field state is initially in a coherent state with average photon number n = |α|2 = 15.Differently from the previous case of initial number state, now entanglement andpurity eventually decay presenting oscillations during the evolutions. We point outthat purity peaks follow entanglement revivals which however does not mean that thelarger the purity (or smaller the mixedness), the larger the entanglement. This canbe immediately seen by comparing, for instance, the behaviors at the time regions2 < gt < 18 (zero entanglement) and 68 < gt < 78 (entanglement revival).

3. Two-qubit Buck-Sukumar model

In this section we consider a variant of the model studied in Sec. 2 which isinspired to the so-called Buck-Sukumar (BS) model [59]. In that work the authorspropose an exactly solvable qubit-field Hamiltonian which is useful to describe


nonlinear interactions. The Hamiltonian for the two-qubit BS model in the spectatorconfiguration is given by

HBS =ω0

2

(σAz + σB

z

)+ νa†a+ g

(a√NσA

+ +√Na†σA

−

), (14)

where N = a†a. Unlike Eq. (1) this model allows an intensity-field dependent coupling.In the resonant case the time evolution operator in the interaction picture is

UBS(t) =

(cos [gt(N + 1)] −iV sin [gtN)]

−iV † sin [gt(N + 1)] cos [gtN ]

)⊗ 1B . (15)

Using the same initial condition for the two-qubit system Eq. (4), the matrix elementsfor the reduced density operator are analogous to Eqs. (6) and (13) (except for thesquare root in the trigonometric functions argument, i.e.

√x → x) for the field in a

number and coherent state respectively.Purity and concurrence for the Bell pair (x=1, φ=π/4) with the field starting in

the number state |n〉 are

P (t) =1

2+

1

8

(4 cos2 [gtn] cos2 [gt(n+ 1)]− 1

)+

1

16(cos [4gtn] + cos [4gt(n+ 1)]) ,

(16)

C(t) = 2 max{0, 1

2(|cos [gtn] cos [gt(n+ 1)]| − |sin [gtn] sin [gt(n+ 1)]|)}. (17)

In Fig. 3(a) we have plotted Eqs. (16) and (17) as functions of time with n = 10photons. A behaviour similar to that of Fig. 3(a) is found between an isolated atomand a Jaynes-Cummings atom [56].

On the other hand, when the field is initially in a coherent state analyticalexpressions for P (t) and C(t) in the two-qubit BS model are cumbersome, as pointedout in the previous section, and we limit to report their plots. Evolutions of purity andconcurrence for this case are displayed in Fig. 3(b) as functions of scaled time for x = 1,φ = π/4 and n = 10. We highlight that now, in contrast to what happened in the JCmodel with an initial coherent field state (see Fig. 2), a complete spontaneous recoveryof the initial entanglement can be found due to the nonlinear atom-field interaction.Purity and entanglement again show the same qualitative behavior but now largervalues of purities always correspond to larger values of entanglement (P = 1/2 whenC = 0 in the plateaux and P = 1 when C = 1 in the peak).

4. Two-qubit spin-boson model

The two-qubit spin-boson model describes two spin 1/2 particles coupled to anenvironment of M non-interacting quantum harmonic oscillators [1], which can beexperimentally realized in cavity and circuit QED [6, 11] and also simulated by all-optical setups with Sagnac interferometers [29, 55]. The pure-dephasing Hamiltonianin the spectator scheme is given by

HSB =ω0

2

(σAz + σB

z

)+

M∑

j=1

ωja†jaj + σA

z ⊗M∑

j=1

(gja†j + g∗j aj

). (18)

Notice that the qubit-environment linear coupling term is an energy conservinginteraction since the central system Hamiltonian commutes with HSB. The


0 π 2πgt

0

0.2

0.4

0.6

0.8

1.0

P(t

)C

(t)

(a)

0 π 2πgt

0

0.2

0.4

0.6

0.8

1.0

P(t

)C

(t)

(b)

Figure 3: Purity (upper red line) and concurrence (lower dark blue line) evolutionfor the spectator two-qubit BS model. The two qubits start in a Bell state. Thefield starts in: (a) number state with n = 10 and (b) coherent state with n = 10. Asequence of entanglement dark periods and complete entanglement recoveries occur inboth cases due to the nonlinear interaction.

corresponding time evolution operator in the interaction picture is

USB(t) =

(∏j D(λj(t)) 0

0∏

j D(−λj(t))

)⊗ 1B , (19)

where D (λj(t)) ≡ exp[(λj(t)a†j−λ∗j (t)aj)] is the usual Glauber displacement operator

for each mode and λj(t) ≡ (gj/ωj)[1 − exp(iωjt)]. If we set all the oscillators in the

ground state ρf (0)=⊗M

j=1 |0〉j j〈0| and the two-qubit system in %Ψ(0), the non-zero


matrix elements of the total density operator in the atomic basis are

ρ11(t) =(1− x

4+ x sin2 φ

)∏

j

|λj(t)〉〈λj(t)|, ρ33(t) =1− x

4

∏

j

| − λj(t)〉〈−λj(t)|,

ρ44(t) =(1− x

4+ x cos2 φ

)∏

j

| − λj(t)〉〈−λj(t)|, ρ22(t) =1− x

4

∏

j

|λj(t)〉〈λj(t)|,

ρ14(t) = ρ14(t)∗ = x sinφ cosφ∏

j

|λj(t)〉〈−λj(t)|, (20)

where |λj(t)〉≡D(λj(t))|0〉j is the coherent state for the j-th oscillator. Aswe have done in previous sections, we trace out over the environmentin order to obtain the reduced density operator of the central system:%11(t)=(1−x)/4+x sin2 φ, %22(t)=%33(t)=(1−x)/4, %44(t)=(1−x)/4+x cos2 φ and%14(t)=%41(t)=x sinφ cosφ exp[−Γ(t)], where Γ(t)=

∑j 4|gj |2(1− cos(ωit))/ω

2j is the

decoherence factor. From the Eqs. (7) and (9) it is trivial to obtain purityand concurrence for the central system. For instance, the explicit expression forconcurrence is

C(t) = max{

0, x| sin 2φ|e−Γ(t) − (1− x)/2}. (21)

For x=1 purity and concurrence are related via

C(t) =

√2P (t)− 2(sin4 φ+ cos4 φ), (22)

which is a generalized form of the expression describing a dephasing process inducedby a local operation acting on a Bell state given by C =

√2P − 1. From Eq. (21),

one finds that entanglement vanishes whenever Γ(t) = − ln((1 − x)/(2x| sin 2φ|)).Assuming all the modes to be identical (gj=g, ωj=ω), with φ=π/4, the time when

entanglement disappears is td = arccos[1 + 1

Mω2

4|g|2 ln((1 − x)/(2x))]. In Fig. 4 we

plot concurrence of Eq. (21) as a function of time for several realizations of gj and ωj

which are randomly chosen from interval [0, 1]. M stands for different dimensions of theenvironment. We emphasize that this time behavior is non-Markovian meaning thatits decay is not exponential at short times, a situation reminiscent of pure-dephasingevolution in the solid state due to inhomogeneous broadening [60, 61].

5. Discussions

In this section we discuss the results for the evolution of purity and concurrence forthe different models studied in previous sections.

5.1. General aspects on the time behaviors

For the two-qubit TC model with the field starting in the vacuum state, concurrence(and also purity) is a periodic function of time as can be seen in Fig. 1. We haveexplored two different initial conditions for the two-qubit system (4): pure entangledstate Fig. 1 (a) and entangled state with a degree of mixedness Fig. 1(b). In both caseswe observe the expected decay of correlations at short times in the initial entangledstate due to the interaction with the field. Fig. 1(a) shows complete entanglementrevivals at times given by gt = nπ. A similar behaviour is shown in Fig. 1(b) butin this case the entanglement remains zero for finite intervals of time, identified


0 π/2 π

t

0

0.2

0.4

0.6

0.8

1.0

Conc

urre

nce

M=5

M=10

M=100

10−2 10−1 100

t

10−1

100

Figure 4: Entanglement evolution for the spectator two-qubit SB model. An ensembleaverage over 106 samples was realized using Eq. (21), x=1, φ=π/4, gj and ωj werechosen randomly from the interval [0, 1], with M= 5 (black), 10 (red), 100 (blue).The inset shows in log scale the Gaussian (exponential) entanglement decay for short(long) times.

as entanglement dark periods [7], followed by complete entanglement recoveries astime goes by. In the case under consideration, the TC interaction permits onlyzero photons or one photon to reside in the cavity, i.e., the cavity acts effectivelyas a two-level system, so the Hilbert space available for the environment is finiteand gives rise to entanglement rebirths in the central system. When entanglementcompletely disappears in the central system, quantum correlations must be containedin other bi-partitions [62, 63], for instance between the isolated qubit and the field orthe central system and the field. This effective three-qubit system is a convenientframework for understanding the dynamical mechanisms of entanglement sharingamong the parts of a composite system with a quantum reservoir [7, 35]. Dynamicalbehaviors qualitatively similar to those obtained in the case when both qubits are open[23, 7] have been here found. This implies that the spectator configuration is able toreproduce general dynamical features exhibited by more complex systems, providedthat each qubit of the system is locally interacting with its own environment.

Concerning the second initial condition for the environmental state in the TCmodel, i.e. the field prepared in a coherent state, we notice that this is the situation inwhich the Hilbert space is formed by an infinite basis of number states. In principle itis possible that entanglement can be shared in arbitrary multipartitions of the Hilbertspace not allowing the complete backflow of information to the central system. Thissort of local coherent-state control leads to revivals of entanglement whose amplitudeeventually decays, as predicted for the case of two open qubits [24, 25]. Purity andconcurrence evolution for the central system have been plotted in Fig. 2 when theaverage photon number of the field coherent state is n = 15. Both quantities oscillatebut the periodicity in both quantities is no longer maintained. This time behaviourresembles the evolution of the atomic inversion in the standard one-qubit JC modelwhere non-complete revivals are consequence of constructive quantum interferencebetween states in the Fock basis [12]. Since we have used the spectator configurationit is easy to see that the time of entanglement revival is given by gtr ≈ 2π

√n.


Successively, considering the intensity-dependent field interaction described by thetwo-qubit BS Hamiltonian in Eq. (14), we have plotted purity and concurrence withthe field in a number (n = 10) and a coherent state (n = 15) in Figs. 3(a) and 3(b),respectively. In contrast to what was observed for the TC model, C(t) and P (t) arenow π-periodic functions independent of the number of photons. Interestingly, whenthe radiation field is initially in a coherent state there are complete entanglementrevivals (see Fig. 3(b)) regardless that we are dealing with an infinite number ofavailable states associated to the coherent field.

In Fig. 4 we have finally shown the entanglement evolution for the two-qubitSB model. Aiming at revealing general features of entanglement deterioration in thissystem, we have performed an ensemble average over 106 samples applying Eq. (21)with x = 1, φ = π/4 and random values of gj and ωj taken from interval [0, 1]. Aswe see, increasing the environmental modes results in a faster decay of entanglement.As expected, for short (long) times a Gaussian (exponential) behaviour is observed[1]. Due to both the initial Bell state of the central system and the dephasing localinteraction, there is no entanglement sudden death, as we can deduce from Eq. (21).

5.2. Concurrence-Purity diagram

A useful way to characterize bipartite quantum states is given by the concurrence-purity diagram or C-P plane [46]. In Fig. 5 we show for convenience a typicalconcurrence-purity diagram specifying the relevant regions. A point on this diagramgives the value of mixedness and entanglement at the same time. Those quantum statesfor which a definite value of purity can reach the maximum degree of entanglementare known as maximally entangled mixed states (MEMS) [64]. MEMS are representedby curve 1 (CMEMS) in the C-P plane. The area below the MEMS curve specifies theregion of physical quantum states. Werner states (φ = π/4 in Eq. (4)) are depictedby curve 2 (CW ). Curve 3 (CD) is given by Eq. (22) with φ = π/4 which correspondsto a decoherence process induced by a dephasing interaction.

In light of the dynamical results we have obtained for purity and concurrence,we analyze their relation using the C-P diagram. We first make some remarks aboutthe nature of the quantum operations involved in our models. We emphasize that thespectator configuration is a physical example of a local quantum operation (channel)acting on a bipartite quantum state (the state of the two-qubit central system). Inthis sense, environment performs operations (trough the interaction) on one of the twoqubits. These local operations can be unital or non-unital. Unital channels are mapsthat leave invariant the uniform state, i.e., the total mixture state. It is known [46]that initial Bell states under the action of unital channels lie in the region bounded bycurves 2 and 3 in the C-P plane (blue shadow) of Fig. 5. Characterizing the behaviourof our quantum channels within this diagram is therefore desirable and can providenew overall insights on concurrence-purity dynamical relations.

In Fig. 6 (a) we show the behaviour of the channel acting on a Bell state generatedby the two-qubit TC dynamics in the spectator scheme. The starting point is theright upper corner in the plane. Two representative cases for the initial state ofthe environment are shown: i) n = 0 and ii) n = 5. For the vacuum state asimple analytical relation between purity and concurrence can be obtained Eq. (12):C = 4

√2P − 1 (red line), which for a long interval of time is outside of the unital region.

This channel is related to the homogenization process describing exponential decay ofcorrelations in which the vacuum state is the fixed point of the dynamics. The case


0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Purity

1.0

0.8

0.6

0.4

0.2

0

Con

curr

ence

1

2

31. CMEMS

2. CW3. CD

Figure 5: C-P plane for two qubits. Curve 1 corresponds to maximally entangledmixed states (MEMS). Curve 2 is for Werner states. The area coloured in blue is theregion allowed for maximally entangled pure two-qubit states when they are under theaction of a unital quantum local channel. This region has a lower and upper boundgiven by CD and CW respectively [46].

n = 5 is shown in blue and gives rise to a rich loop structure due to immeasurabilityand non-Markovian behaviour in the evolution of purity and concurrence. It must bementioned that the associated C-P line for the vacuum state is also a loop over itselfreaching zero entanglement at times gt = (m + 1/2)π. These loops are exceptionsto the rule that lines in the C-P plane must be non-increasing if they are generatedby Markovian semigroup dynamics. Hence, the appearance of this loops is due tothe non-Markovian evolution considered in this work as we were able to obtain theexact reduced density operator for the central system. It should be noted that similarresults (not shown) for the Buck-Sukumar interaction in the C-P diagram can beobtained; in contrast to the spectator two-qubit TC model, closed loops emerge dueto the π-periodicity in the purity and concurrence.

At this point it is interesting to see the C-P dynamics for an initial coherentstate for the environment using the results of subsection 2.2. For an average numberof field excitation n = 100, signatures of long-time entanglement revivals are obtainedbefore their occurrence (see Fig. 6(b)). Almost all the action of the local operationis contained in the unital region except for a small part generated by the short timedynamics near to the upper right corner. The corresponding C-P representation forthe spectator two-qubit SB dynamics is also shown in Fig. 6(b) (dashed line) usingthe obtained generalized expression in Eq. (22) with φ = π/6. As expected we observea typical decoherence process induced by dephasing, this process being represented asa rescaled CD curve.

5.3. Operational use of the concurrence-purity relations

We now briefly discuss on the possible usefulness to have quantitative relationsbetween concurrence and purity for implementing some specific protocols. It is knownthat entanglement must overcome some quantitative thresholds, for a given valueof state purity, in order to allow quantum processes, such as teleportation [49, 50],


0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Purity

1.0

0.8

0.6

0.4

0.2

0

Con

curr

ence

n = 5

n = 0, C = 4√

2P − 1

(a)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Purity

1.0

0.8

0.6

0.4

0.2

0

Con

curr

ence

n = 100

C =√

2P − 5/4

(b)

Figure 6: C-P plane representation for the local operation induced by Tavis-Cummingsinteraction. The central system starts in a Bell state and the field in: (a) Fock staten=5 (blue lines) and n=0 (red line), (b) coherent state n=100 (red line). Dashed linein (b) corresponds to a dephasing channel generated by the SB interaction (see text).

entanglement swapping [51] and entanglement percolation [52]. Our results underspecific dynamical conditions allow to only measure purity of the system state at agiven time t for obtaining the value of concurrence and then checking if it is sufficientfor the desired task. Such a procedure will in turn provide the time regions withinwhich the task can be performed.

We focus on the recently reported concurrence threshold for entanglementnecessary to realize a teleportation protocol with quantum speedup [50]. Such athreshold is equal to Cth = (

√ρ22 − √ρ33)2 in the case when the entangled state

shared between the two parties is a X state, which is just the one we have duringthe evolutions here considered. For instance, for the SB dephasing model, whereρ22(t) = ρ33(t), one immediately gets Cth = 0 at any time. The system state can bethus exploited for teleportation until C(t) > 0 = Cth, which in turn means wheneverpurity is above its minimum value P (t) > Pth ≡ sin4 φ+ cos4 φ (see Eq. (22)). For the


plot of Fig. 6(b) it must be P (t) > 5/8. Instead, for the TC model with the vacuumfield state and the two qubits initially prepared in a Bell state, the entanglementthreshold is time-dependent, namely Cth(t) = (1/2) sin2(gt). Quantum teleportationis then achievable at those times such that C(t) = | cos(gt)| > Cth(t), which in termsof state purity also means P (t) > Pth(t) ≡ [1 + C4

th(t)]/2 according to Eq. (12).

6. Conclusions

In this paper we have presented different exactly solvable models for the dynamicsof entanglement and purity of a simple two-qubit central system. We have takenadvantage of the spectator configuration, where a qubit is isolated, in order torealize a single local quantum operation acting on a maximally entangled pure state.Furthermore, it allows for straightforwardly find the evolved two-qubit density matrixonce the quantum map of the open qubit is known. We have obtained explicitanalytical expressions for purity, concurrence and their dynamical relations (Eqs. 12and 22) using Tavis-Cummings, Buck-Sukumar and spin-boson type interactions. Ourresults confirm that even in the spectator scheme the entanglement can disappearat a finite time depending on the initial conditions, as previously found in otheropen quantum systems [23, 32, 41]. Long-time entanglement revivals appear when acoherent state of the radiation environment is considered, showing that even simplersystems that the ones treated in previous works [3, 7, 24, 25] can reveal generalfeatures of entanglement evolution. In fact, the qualitative behaviors of the dynamicsof quantum correlations, like entanglement, are analogous for bipartite systems of bothopen qubits and only one open qubit provided that the qubits are independent andlocally interacting with their own environment.

As a further source of information we have exploited the C-P diagram tocharacterize how local actions ruled by the environment affect an initial two-qubitBell state. For the TC and BS interactions, the two-qubit state can reach pointsoutside of the unital region which thus evidences the non-unital nature of these kindof quantum maps commonly employed in the context of quantum optics. We have alsodiscussed the potentiality of having concurrence-purity dynamical relations to assessquantitative entanglement and purity thresholds at a given time which allow specificquantum tasks, such as teleportation.

These results motivate further studies of dynamical characterization of thresholdsof purity and entanglement for implementing processes like entanglement swapping [51]and entanglement percolation [52]. For future works, it would be also interesting toconsider more realistic models in the spirit of the spectator configuration, for instanceintroducing spontaneous emission and cavity photon losses by means of Lindbladmaster equations.

Acknowledgements

CGG, RRA and DE would like to express their gratitude to CONACyT for financialsupport under scholarships No. 385108, 379732 and 413926, respectively.


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